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Engineering
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Let a, b, c be such that  b(a+c) \neq 0.If

then the value of n is

  • Option 1)

    any even integer

  • Option 2)

    any odd integer

  • Option 3)

    any integer

  • Option 4)

    zero

 

As we learnt in 

Value of determinants of order 3 -

-

 

 \begin{vmatrix} a &a+1&a-1 \\ -b & b+1 &b-1\\ c&c-1& c+1 \end{vmatrix}+\begin{vmatrix} a+1&b+1&c-1 \\ a-1& b-1 &c+1\\ \left ( -1\right )^{n+2}a& \left ( -1\right )^{n+1}b& \left ( -1\right )^{n}c \end{vmatrix}= 0

\begin{vmatrix} a &a+1&a-1 \\ -b & b+1 &b-1\\ c&c-1& c+1 \end{vmatrix}+\begin{vmatrix} a+1&b+1&c-1 \\ a-1& b-1 &c+1\\ \left ( -1\right )^{n}a& \left ( -1\right )^{n}b& \left ( -1\right )^{n}c \end{vmatrix}= 0

\begin{vmatrix} a &a+1&a-1 \\ -b & b+1 &b-1\\ c&c-1& c+1 \end{vmatrix}+(-1)^{n}\begin{vmatrix} a+1&b+1&c-1 \\ a-1& b-1 &c+1\\ a& -b& c \end{vmatrix}= 0

C_{1}\leftrightarrow C_{2}

\begin{vmatrix} a+1 &a&a-1 \\ b+1 & -b &b-1\\ c-1&c& c+1 \end{vmatrix}+(-1)^{n}\begin{vmatrix} a+1&b+1&c-1 \\ a-1& b-1 &c+1\\ a& -b& c \end{vmatrix}= 0

C_{2}\leftrightarrow C_{3}

\begin{vmatrix} a+1 &a-1&a \\ b+1 & b-1 &-b\\ c-1&c+1& c \end{vmatrix}+(-1)^{n}\begin{vmatrix} a+1&b+1&c-1 \\ a-1& b-1 &c+1\\ a& -b& c \end{vmatrix}= 0

\begin{vmatrix} a+1 &a-1&a \\ b+1 & b-1 &-b\\ c-1&c+1& c \end{vmatrix}\ (1+(-1)^{n})=0

\therefore 1+(-1)^{n}=0

So n is any odd integer.


Option 1)

any even integer

Incorrect Option

 

Option 2)

any odd integer

Correct Option

 

Option 3)

any integer

Incorrect Option

 

Option 4)

zero

Incorrect Option

 

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Engineering
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Directions : Questions are Assertion- Reason type questions. Each of these questions contains two statements :

Statement- 1 (Assertion) and Statement - 2 (Reason).

Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice.

Question : Let A be a  2 x 2 matrix

Statement-1 : adj\, (adj\; A)=A

Statement-2 : \left |adj\; A \right |=\left | A \right |

 

 

  • Option 1)

    Statement-1 is true,Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

  • Option 2)

    Statement- 1 is true, Statement-2 is false

  • Option 3)

    Statement-1 is false, Statement-2 is true

  • Option 4)

    Statement-1 is true, Statement-2 is true; Statement-2 is correct explanation for Statement-1

 

As we learnt in 

Property of adjoint of A -

\left | adj A \right |=\left | A \right |^{n-1}  

- wherein

adj A denotes adjoint of A and  \left |A \right |  denotes determinant  of A and n is the order of the matrix

 

 |adjA|= |A|^{n-1}= |A|^{2-1}= |A|      Statement 2 is true

adj|adjA|= |A|^{n-2}A= |A|^{0}A= A  Statement 1 is true

 


Option 1)

Statement-1 is true,Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

Correct Option

Option 2)

Statement- 1 is true, Statement-2 is false

Incorrect Option

 

Option 3)

Statement-1 is false, Statement-2 is true

Incorrect Option

 

Option 4)

Statement-1 is true, Statement-2 is true; Statement-2 is correct explanation for Statement-1

Incorrect Option

 

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